Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumption that 0 and ∞ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm-Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.Keywords: Krein space, non-negative operator, definitizable operator, critical point, nonreal spectrum, spectral points of positive and negative type, indefinite weight, singular Sturm-Liouville equation, elliptic differential operator MSC 2010: 46C20, 47B50, 34B24, 47F05, 34L15
The number of negative squares of all self-adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a Krein space is characterized in terms of the behaviour of an abstract Titchmarsh-Weyl function near 0 and ∞. These results are applied to a large class of symmetric and self-adjoint indefinite Sturm-Liouville operators with indefinite weight functions.
Abstract. We study a Sturm-Liouville expression with indefinite weight of the form sgn(−d 2 /dx 2 + V ) on R and the non-real eigenvalues of an associated selfadjoint operator in a Krein space. For real-valued potentials V with a certain behaviour at ±∞ we prove that there are no real eigenvalues and that the number of non-real eigenvalues (counting multiplicities) coincides with the number of negative eigenvalues of the selfadjoint operator associated toThe general results are illustrated with examples.
We consider a singular Sturm-Liouville expression with the indefinite weight sgn x. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of ∞. Moreover, in this situation, the point ∞ is a regular critical point. We construct an operator A = (sgn x)(−d 2 /dx 2 + q) with non-real spectrum accumulating to a real point. The obtained results are applied to several classes of Sturm-Liouville operators.
Second order equations of the formz(t) + A 0 z(t) + Dż(t) = 0 are considered. Such equations are often used as a model for transverse motions of thin beams in the presence of damping. We derive various properties of the operator matrix A = 0 I −A 0 −D associated with the second order problem above. We develop sufficient conditions for analyticity of the associated semigroup and for the existence of a Riesz basis consisting of eigenvectors and associated vectors of A in the phase space.
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